The Analytic Continuation of the

OverviewVery high level overview because of the amount of materialin multiple branches of mathematics.

Complex Systems A New Kind of Science

ArithmeticDynamics of the Complex PlaneCombinatorics

New Kind of Science

Chaos beyond exponentiation.

Vertical catalog of complex systems.Based on iterated functions.Arithmetic and physics are two major rolesplayed by iterated functions. Iterated functions as a candidate for afundamental dynamical system in bothmathematics and physics.

Dynamics and Combinatorics

Arithmetic Arithmetic is part of the Foundations ofMathematics. Ackermann function is a recursive functionthat isnt primitively recursive. Different definitions of the Ackermannfunction. Transfinite mathematics

Transfinite Ackermann Expansions

a k useful for creating a series of interesting transfinite number.Transfinite nature of circulation:22

= 4

23

= 2 (2 2 ) = 24= 2 (2 (2 2 ) ) = 2 (2 4 ) =

32

= 33= 3 (3 3 ) = 3 (3 (33) ) =

Tetration First objective is understanding tetration. What if tetration and beyond is vital formathematics or physics? With so many levels of self organization inthe world, tetration and beyond likelyexists.

Julia set for the map of ez

Tetration by period

Tetration by escape

Dynamical Systems Iterated function as a dynamical system. Analytic continuation can be reduced to a problemin dynamics. Taylor series of iterated function. Mostmathematicians believe this is not possible, but myresearch is consistent with other similar researchfrom the 1990s. Iterated exponents for single valued and iteratedlogarithms for multi-valued solutions.

Subject is in protomathematics stage, but becoming acceptable areas

of research; numerous postings on sci.math.research lately.Arithmetic Dynamics Combinatorics ArithmeticIf maps are flows, then the Ackermann function is transparentlyextended.Suggests time could behave as if it is continuous regardless of whetherthe underlying physics is discrete or continuous.Continuous iteration connects the old and the new kinds ofscience. Partial differential iterated equationsTetration displays sum of all paths behavior, so logical starting placeto begin looking for tetration in physics is QFT and FPI. Tetration andmany other iterated smooth functions appear compatible with theLorenz transforms and Bells Theorem.